Oval

A closed convex $C^2$-smooth curve in $\R^2$. The points of an oval at which the curvature is extremal are called the vertices of the oval. The number of vertices is at least four.

Let $E$ be an oval, traversed counter-clockwise, in the plane with rectangular Cartesian coordinates $x,y$ let $h$ be the distance from the origin $O$ to the directed tangent line to $E$ ($h>0$ if the rotation of the tangent line relative to $O$ is counter-clockwise). Then the equation of the tangent line is

$$x\cos\tau + y\sin\tau=h(\tau),$$

where $\tau$ is the angle made by the tangent line and the axis $Ox$. The quantity $h(\tau)$ is called the support function of the oval. The radius of curvature of the oval is

$$r=h+\frac{d^2 h}{d\tau^2};$$

and the length of the oval (Cauchy's formula) is

$$L=\int\limits_{-\pi}^\pi h(\tau)d\tau.$$

The following isoperimetric inequality holds for the length $L$ and the area $F$ of the region inside the oval:

$$L^2-4\pi F\geq 0$$

(for more details see Bonnesen inequality).

Comments

Sometimes smoothness is not assumed, so that any closed convex curve in $\R^2$ is called an oval. In finite (projective) geometry the term "oval" denotes a special kind of ovoid.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a3]	S.S. Chern, "Curves and surfaces in Euclidean space" , Prentice-Hall (1967)
[a4]	T. Bonnesen, W. Fenchel, "Theorie der konvexen Körper" , Springer (1934)